Download Our Universe SPA-4101

Our Universe
SPA-4101"
Synopsis This module provides an introduc5on to modern astronomy and astrophysics Topics covered include an introduc5on to the history of the subject (da5ng back to ancient greek astronomers), the Sun, Solar System bodies (planets, satellites/
moons, asteroids, comets, Kuiper belt), extrasolar planets, stars (their forma5on, evolu5on and end-­‐of-­‐
life remnants), galaxies, and cosmology (the history and long-­‐term fate of the universe). The emphasis will always be on how the laws of physics and mathema5cs can help us understand astronomical phenomena Course text Universe (9th edi5on) by Freedman, Geller & Kaufmann will be the main text book used in this module. We will also make use of Astronomy Today (7th edi5on) by Chaisson & McMillan Module web page hNp://qmplus.qmul.ac.uk/course/view.php?id=3094 Contains: Module syllabus Lecture notes Problem sets (available on Wednesday 4pm, due in by 4pm Wednesday of following week) Problem set solu5ons (available 2 days aYer submission deadline) Assessment Final examina5on (April/May 2015) – 80% Eight problem sets – 20% Examina5on 2 hr 30 min Examina5on in April/May Two Sec5ons -­‐ A (50 marks) and B (50 marks) Examina5on Sec5on A (50 marks) About 10 basic ques5ons covering en5re module Answer ALL ques+ons Designed to test at the pass/fail boarder Students should be able to pass the examina5on through Sec5on A alone (i.e., score 40) Examina5on Sec5on B (50 marks) About 4 in-­‐depth, structured, and unseen ques5ons on any selected parts of the module. Answer ONLY TWO ques+ons Ques5ons will contain unseen material Problem sets 8 problem sets: consis5ng of about 4 or 5 ques5ons. Assessed weekly throughout the term (star5ng week 2) Will be published on the module web page on Wednesday of the week preceding the submission deadline. Answers will published on the module web page 2 days aYer the submission deadline. Problem sets Marked scripts will be returned within 1 week of the submission deadline Scripts should be posted in the relevant pigeon hole in the SPA administra5ve area Problem sets LATE SUBMISSION Scripts submiNed late WILL BE accepted and marked only as described below Submi'ed before answer publica4on: Marks stand provided there are sufficient extenua+ng circumstances (which must be beyond the student’s control and supported by evidence – e.g., a medical cer5ficate). Otherwise they will be marked for feedback purposes only – and will score ZERO •  Submi'ed a6er publica4on of the answers: Scripts will not be marked ! Module structure 33 Lecture slots (3 per week): Mon 10-­‐11 Francis BancroY 3.26, Mon 12-­‐13 Francis BancroY 4.04/4.08, Tue 12-­‐13 Geography 2.26 2 exercise classes per week (beginning week 2): Monday 16:00-­‐17:00 (G.O.Jones 208 and LG7) Tuesday 16:00-­‐17:00 (G.O. Jones 208 and LG7) Each student will aNend just one of these sessions No mid-­‐term tests ANendance ATTENDANCE AT ALL LECTURES IS COMPULSORY (5metable clashes excepted) The aNendance requirement is 100 %, and any student whose aNendance average is below 70% WILL be warned. If the record is not rec5fied promptly, or aYer rec5fica5on the student falls below 70% again, they will receive a second and final warning. Any student on 2 warnings who’s aNendance record falls below, or fails to improve towards, 70 % WILL be automa5cally deregistered from the module without further warning. ANendance ATTENDANCE AT ALL LECTURES IS COMPULSORY (5metable clashes excepted) Deregistra5on = comple5on of the module at that point = FAIL with no further lectures or assessments, no examina5on, and no re-­‐sits. Deregistra4on from 3 modules in one year OR from a total of 7 modules during a degree course must result in termina4on of registra4on from the College. Our Universe – in a nutshell The Sun Basic facts: 1.5 x 1011 m from Earth (i.e. 1 Astronomical Unit) Radius = 7 x 108 m (Radius ~ 1/220 of 1 AU) Mass = 1.989 x 1030 kg Tsurface = 5700 K Tcentral = 1.5 x 107 K Energy source is nuclear fusion reac5ons The Sun goes through an 11 year cycle during which the level of surface ac5vity (sun-­‐spots, flares, coronal mass ejec5ons) varies. The cycle is related to the level of magne5c ac5vity, and during 11 years the polarity of the Sun’s magne5c field changes from N to S or vice versa. Solar System Planets Small, rocky planets close to the Sun. Gas giant planets far from the Sun. Dwarf planets in the asteroid belt and Kuiper belt. Rela5ve sizes of the Sun and the planets of our Solar System. Sun is 1000 5mes more massive than Jupiter. Rela5ve sizes of the planets Jupiter is 300 5mes more massive than the Earth Interior to the asteroid belt the Solar system contains 4 rocky planets -­‐ with very diverse proper5es ! The Earth is the only one of the inner planets to host a large moon Venus has a scorching surface temperature of 735 K due to runaway greenhouse effect, and a surface atmospheric pressure of 92 bar Mars is now outside of the habitable zone, but shows evidence of having had a warmer past with liquid water on its surface Between Mars and Jupiter we have the asteroid belt, consis5ng of tens of 1000s of rocky and icy bodies ranging in radius from 100s metres to 500 km Orbital evolu5on of the small body popula5on in the inner solar system Out beyond the asteroid belt the nature of the planets changes drama5cally -­‐ why is this ? Jupiter and Saturn are gas giants composed largely of hydrogen and helium -­‐ but probably with massive rocky and icy cores at their centres Uranus and Neptune are normally referred to as ice giants – they are composed largely of rock and ice with only ~ 10 % of their mass being in a H and He envelope The giant planets host impressive satellite and ring systems with diverse proper5es that are being studied by in-­‐situ spacecraY such as Cassini The outer solar system contains a large popula5on of icy bodies – the Kuiper belt. This acts as a source of short period comets. The Pluto-­‐Charon system is a member of this belt, along with numerous other dwarf planets. The Solar system is surrounded by a quasi-­‐ spherical halo of small bodies – the Oort cloud. These bodies were scaNered onto highly ellip5cal and inclined orbits by the giant planets at the end of Solar system forma5on. The Oort cloud is the source of long-­‐period comets The Solar neighbourhood Moving out of the Solar System we enter the realms of interstellar space and eventually our nearest stellar neighbours – a triple star system consis5ng of Proxima Centauri (shown by the red circle above) which appears to be gravita5onally bound to the close binary system Alpha Centauri AB (the bright star on the leY). The bright star on the right (beta Centauri) is not physically associated with this system. The above diagram (right) shows the stars which are nearest to the Sun. Sirius is the brightest star in the sky and one of the closest to us. As we and our stellar neighbours orbit the centre of the Galaxy the proximity of nearby stars changes (right figure). In approximately 40,000 years Ross 248 will become the nearest star to the solar system. The Sun orbits in the Milky Way Galaxy at approximately 2/3 of the distance from the Galac5c centre. It takes approximately 200 Myr to complete 1 revolu5on. The Sun sits in the plane of our spiral galaxy, so viewed from Earth we see a flat distribu5on of stars and nebulae. The image below shows an IR composite of the Milky Way taken by the COBE satellite, allowing us to see through the dust that obscures our view of the Galac5c centre at visible wavelengths. It is es5mated that the Milky Way contains 1011 stars. Stars and stellar evolu5on A star is a quasi-­‐spherical mass of gas that generates energy through nuclear fusion reac5ons in its core. AYer their forma5on, stars spend most of their lives on the main sequence as they convert hydrogen into helium within their core. Main sequence stars range in mass from 0.1 – 100 Solar masses, with more massive stars having larger radii Stellar evolu5on is ul5mately driven by the conversion of hydrogen to helium, and from helium to heavier elements such as carbon and oxygen. During the laNer stages of their lives, stars leave the main sequence, becoming red giants or supergiants depending on their masses. Stars less massive than approximately 10 solar masses loose their outer envelopes by passing through a planetary nebula phase, leaving behind a white dwarf -­‐ such a fate awaits the Sun in approximately 6x109 years. Stars more massive than 10 solar masses end their lives aYer ~ 10 Myr in a supernova explosion, leaving behind either a neutron Planetary nebula Supernova remnant – star or black hole. Crab nebula Star and planet forma5on Stars form from the gravita5onal collapse of dense interstellar gas clouds. One of the closest star forming regions to the Sun is the Orion nebula – which glows because the hydrogen gas is heated by an embedded cluster of recently formed high mass stars. Close inspec5on of some of the embedded low mass stars reveals that they are surrounded by orbi5ng discs of gas and dust – believed to be loca5ons where new planetary systems are forming Extrasolar planets The first extrasolar planet orbi5ng a Sun-­‐like star was discovered in 1995. Since this 5me more than 1800 exoplanets have been confirmed, and NASA’s Kepler space-­‐craY has detected more than 3000 planetary candidates. These sytems have been detected by a variety of methods: detec5ng the wobble of the star as the planet and star orbit their common centre of mass; detec5ng the dimming of the starlight as the planet passes in front of its host star; direct imaging. We now know of the existence of Jupiter-­‐like, Neptune-­‐like, and Earth-­‐like exoplanets, and planets much more massive than Jupiter. Many of these planetary systems are very different in their orbital structure than the solar system. For example, some planets have orbital periods of three days or less, resul5ng in extreme hea5ng by the host star Extragalac5c astrophysics Our Milky Way Galaxy is just one of billions of galaxies in the observable universe. We are members of a small cluster of galaxies called the Local Group whose largest members are Andromeda, the Milky Way and The Pinwheel M33. Galaxies come in all shapes and sizes: spirals, giant ellip5cals, irregulars, dwarf spheroidals,… The Local Group is not isolated. It is in gravita5onal interac5on with nearby Groups including the Virgo Cluster. This is a giant cluster of 2,000 galaxies situated ~ 60 million light years away. It is the center of an immense group of nearby galaxies known as the Local Supercluster, of which the Milky is a member. The Virgo Cluster influences all nearby galaxies and groups via gravity. The Virgo Cluster is merely a minor member of a much larger whole. The neighbouring superclusters are 3-­‐4 5mes larger than the Virgo Cluster, spanning a distance of ~ 100 million light years. These in turn form massive chains and walls that weave throughout the cosmos. Big Bang Cosmology In 1929 Edwin Hubble ploNed the recession velocity versus distance for a collec5on of galaxies, and showed that more distant galaxies are receding faster. This implies that the Universe is undergoing uniform expansion. The Universe must have therefore been much smaller in the past, giving rise to the concept that the Universe began with the Big Bang. Observa5on of the cosmic microwave background radia5on confirms that the Universe began with a hot Big Bang. Observa5ons of distant supernovae suggest that the Universe is actually expanding at an accelera5ng rate. The origin of this is unknown, but has led to the concept of Dark Energy. The fate of our Universe appears to be endless expansion at an ever increasing rate… Lecture 2 Astronomical consequences of Earth’s spin Viewed from the Earth’s surface, all celes5al objects (Sun, Moon, planets, stars) appear to rotate across the sky. Viewed from a fixed loca5on, the Sun returns to the same posi5on in the sky rela5ve to a fixed point on the horizon aYer 24 hours. In the northern hemisphere it appears that the sky rotates about a fixed point very close to the star ‘Polaris’. At the present 5me the Earth’s rota5on axis points in a direc5on very close to Polaris -­‐ the North star. We know that this ‘diurnal’ (daily) mo5on of the Sun, Moon and stars is due to the rota5on of the Earth. At all 5mes (except during a solar eclipse) exactly half of the Earth is illuminated by the Sun, while the other half is in darkness. On a clear night we see that the stars form paNerns in the sky – ‘constella5ons’. Rota5on of the Earth causes these constella5ons to change their apparent posi5on in the sky. There are 88 constella5ons. Astronomers have divided the night sky into a patchwork of 88 regions each named aYer the constella5on contained within it. Each constella5on contains stars with no close physical rela5onship in 3D space – the apparent paNerns arise en5rely due to the loca5on from which we observe the stars and their 2D projec5on on the sky. We use the constella5on of Orion (‘the hunter’) to illustrate these basic points. At different 5mes of the year different constella5ons become visible or disappear from view because of the Earth’s orbit around the Sun. At mid-­‐la5tudes in the northern hemisphere (e.g. London) Cygnus is overhead at midnight in July, Andromeda is overhead in September, and Perseus is overhead in November. The celes5al sphere When considering the posi5ons of celes5al bodies on the sky it is useful to define the concept of the celes5al sphere. Some ancient socie5es believed that the stars consisted of fires located on the inner surface of a gigan5c hollow sphere – the celes5al sphere. Although our view of the cosmos has changed immeasurably, it is s5ll convenient when mapping the posi5ons of celes5al bodies to imagine that they are located on the inner surface of the celes5al sphere. The celes5al sphere has the Earth at its centre. Its north and south poles coincide with the direc5ons of the Earth’s north and south poles (defined in turn by the Earth’s spin axis). The star Polaris is less than 1o away from the celes5al north pole. The plane of the celes5al equator coincides with the Earth’s equatorial plane. The point in the sky directly overhead of an observer is the ‘zenith’. The diagram on the right shows an observer located at la5tude 35o north with the zenith clearly marked. Note that the celes5al sphere has been 5lted in this diagram so that its north pole is to the right and its south pole is to the leY. The celes5al equator is not marked. The spin of the Earth from west to east makes it appear that the celes5al sphere rotates about its axis from east to west. Celes5al bodies that are located below the horizon of the celes5al sphere are not visible to the observer, and those above the horizon Note that this is the horizon for the are (weather permiwng !). observer and not the celes5al equator Stars located within 35o of the celes5al north pole never set for this observer, because they no not venture beneath the horizon. Those located within 35o of the celes5al south pole never rise above the horizon and are therefore not visible to the observer. For most observers at mid-­‐la5tudes on the Earth, the stars rise and set at an angle to the horizon (as shown in the leY diagram below). At either the north or south pole, however, the paths of the stars are parallel to the horizon due to the rota5on of the celes5al sphere (or equivalently the spin of the Earth). At the equator, stars follow trajectories that are perpendicular to the horizon due to the apparent rota5on of the celes5al sphere. Celes5al coordinates By analogy with longitude and la5tude on the Earth, astronomers use a coordinate system to define the posi5ons of celes5al objects on the celes5al sphere. The posi5on of a star, for example, is defined by two angles: right ascension and declina5on shown in the diagram. Right ascension (RA) is measured eastward from a line that runs between the north and south poles through a posi5on on the celes5al sphere called the vernal equinox. The Sun is located at the vernal equinox on 21st March (first day of spring). RA is therefore analogous to longitude on the Earth. Instead of using degrees or radians to measure RA, the units hours, minutes and seconds are used. Given that it takes 24 hours for the celes5al sphere to rotate once through 360o, 1 hour corresponds to 15o, 1 minute corresponds to 15o/60, and 1 second corresponds to 15o/(3600). Declina5on (Dec) is the angle measured from the celes5al equator, and is analogous to la5tude on the Earth. Dec is measured in units of degrees, arcminutes and arcseconds. The direc5on in which the Earth’s rota5on axis points shiYs slowly with 5me due to gravita5onal interac5on with the Sun and Moon – this is called precession of the equinoxes (see later). This means that the north and south poles of the celes5al sphere also shiY slowly rela5ve to the stars, causing their RA and Dec values to change. Therefore astronomical catalogues need to be updated on a regular basis and a par5cular value of RA and Dec only applies for a short period of 5me. For example, the celes5al coordinates of Rigel, a bright star in Orion, in the year 2000 were given as: RA = α = 5h 14m 32.2s Dec = δ = -­‐8º 12ʹ′ 06ʹ′
Note that the symbol α oYen denotes RA, and δ denotes Dec. A nega5ve value of δ signifies that the star is in the southern hemisphere. It is important to note that as the celes5al sphere rotates during the night, stars that arrive at a fixed point in the sky for an observer (e.g. the zenith) have the same value of δ but different values of α because of the way in which these angles are defined (see diagram on previous page to understand this).
Orbital mo5on of the Earth The orbit of the Earth around the Sun, combined with the 5lt of the Earth’s spin axis by 23.5o rela5ve to the orbital angular momentum vector (normal to the orbital plane), leads to changing of the seasons. The important point to note is that the conserva5on of spin angular momentum causes the Earth’s spin axis to point in a constant direc5on as it orbits around the Sun (ignoring the precession of the equinoxes for the 5me being). On 21st June the projec5on of the Earth’s spin axis onto the orbital plane coincides with the line joining the centres of the Sun and Earth, with the north pole poin5ng in its most sunward direc5on. One effect of this is to make the hours of daylight in the northern hemisphere significantly longer than the hours of darkness, increasing the warming effect of the Sun rela5ve to the southern hemisphere. During the summer months the Sun is also higher in the sky such that a beam of radia5on incident on the ground is concentrated over a smaller area than when the Sun is lower in the sky. For example, on 21st June the Sun is directly overhead at mid-­‐day at the Tropic of Cancer (la5tude 23.5o north), such that solar radia5on is incident at an angle of 90o to the Earth’s surface. The longer daylight hours combined with the steeper angle of incidence for solar radia5on explains why it is warmer in summer and colder in winter. The above diagram shows how the Sun’s radia5on is incident on the Earth during the winter and summer sols5ces (21st of December and June, respec5vely). Mo5on of the Sun on the celes5al sphere The orbit of the Earth around the Sun causes the Sun’s apparent posi5on on the celes5al sphere to move on a circle, comple5ng one revolu5on in a year. The apparent mo5on of the Sun is from west to east because of the direc5on of the Earth’s orbital mo5on. This is the opposite direc5on to the diurnal mo5on of celes5al bodies across the sky that arises because of the Earth’s spin. Because the equatorial plane of the celes5al sphere coincides with the Earth’s equator, and the Earth’s orbital plane is inclined by 23.5o to the equator, the path followed by the Sun throughout the year is 5lted by 23.5o to the equatorial plane. The path of the Sun across the sky is called the eclip5c. The vernal and autumnal equinoxes occur when the Sun crosses the equatorial plane of the celes5al sphere on 21st March and September. Equinox means ‘equal night’ in La5n -­‐ day and night are of equal length (12 hours) during the equinoxes. The summer sols5ce occurs when the Sun is at its most northerly point on the celes5al sphere, such that the northern hemisphere is bathed in maximum levels of sunlight. The winter sols5ce occurs when the Sun is at its most southerly point. Sun’s daily path across the sky The Sun rises exactly in the east and sets in the west on the equinoxes. In the summer (northern hemisphere) the Sun rises in the north-­‐east and sets in the north-­‐west. In the winter the Sun rises in the south-­‐east and sets in the south-­‐west. For la5tudes north of 23.5o N the Sun never reaches the zenith. The zodiac is the set of constella5ons of the celes5al sphere that coincide with the eclip5c. During the Sun’s apparent mo5on around the celes5al sphere it passes through the 12 constella5ons of the zodiac. This is also true for the planets in our Solar System and the Moon. Their orbits are only modestly inclined rela5ve to the Earth’s, so they follow paths around the celes5al sphere that remain close to the eclip5c At the vernal equinox on 21st March the Sun is in the constella5on Pisces. Precession of the equinoxes The gravita5onal fields of the Sun and Moon ac5ng on the rota5ng Earth cause the axis of rota5on to precess like a spinning top or gyroscope. The precession period is 26,000 years. This effect is en5rely due to the fact that the Earth is not spherical, but is an oblate spheroid (its spin causes modest rota5onal flaNening). The equinoxes occur when the line joining the Earth and Sun forms a right angle with the Earth’s spin axis. In the absence of precession this would occur exactly twice per year. Precession of the spin axis causes the equinoxes to occur earlier by 1/26,000 of a year. Over 5me this causes the seasons to become out of synch with the calendar without an appropriate correc5on. Consequences of precession The equinoxes occur when the line joining the Earth and Sun forms a right angle with the Earth’s spin axis. In the absence of precession vernal equinoxes would occur exactly one year apart. Precession of the spin axis causes the equinoxes to occur earlier by 1/26,000 of a year. Astronomers refer to the 5me required for the Earth to orbit exactly once round the Sun the sidereal year = 365.2565 days. The 5me between successive vernal equinoxes is referred to as the Tropical year = 365.2422 days. Over 5me precession causes the seasons to shiY. For example Orion is currently a winter constella5on in the northern hemisphere, but will become a summer one in 13,000 years. Similarly, the con5nuous shiY of the equinoxes would cause the first day of spring to become earlier than 21st March without compensa5on in the calendar. The Gregorian calendar used today corrects for precession, ensuring that the equinox always occurs on 21st March. Finally, precession changes the direc5on in which the Earth’s spin axis points. At the present 5me the spin axis points toward the star Polaris in the northern hemisphere. The diagram above shows that on a 5me scale of a few thousand years precession will cause this star to no longer be the Pole star. Lecture 3 The Earth and Moon The Moon gives rise to a number of important phenomena: ocean 5des, lunar and solar eclipses. It’s journey through different phases (new moon, quarter moon, full moon, etc) over a period of approximately four weeks forms the basis of the modern calendar. Basic facts: The Moon orbits the Earth once every 27.3 days The Moon shows the same face toward the Earth throughout its orbit. The Moon’s spin period therefore equals its orbital period -­‐ we call this synchronous rota5on. The plane of the Moon’s orbit around the Earth is 5lted by 5.2o with respect to the orbital plane of the Earth around the Sun (i.e the eclip5c). The Moon’s orbit around the Earth is modestly ellip5cal so its distance from the Earth is not constant. Lunar phases Except during lunar eclipses (see later), exactly half of the Moon’s surface is illuminated by the Sun. As it the orbits the Earth the frac5on of the illuminated surface that we observe changes, depending on the rela5ve posi5ons of Earth, Moon and Sun. The diagram to the right explains the origin of the different lunar phases and also gives the names of the different phases (New moon, waxing gibbous, etc…) Although the 5me for the Moon to complete one orbit around the Earth is 27.3 days, the 5me between successive New Moons in 29.5 days. As the diagram shows, this is because during one lunar orbit the Earth moves around the Sun on its orbit by approximately 30o. The 5me for one orbit (rela5ve to the fixed stars) is called a sidereal month. The 5me between successive New Moons is called a synodic month. Lunar and Solar eclipses Lunar and Solar eclipses only occur when the Sun, Moon and Earth lie in a straight line. For this to occur the Moon needs to be full (for a lunar eclipse) or new (for a solar eclipse) and needs to be at one of the points where the 5lted lunar orbit crosses the plane of the Earth’s orbit (the eclip5c – which is where the name comes from !). As shown in the diagram, the two points where the Moon crosses the eclip5c are called nodes, and the line joining these is called the line of nodes. Unfavourable for eclipse The above diagram shows a scenario in which the New or Full Moon does not occur at one of the nodes (top), crea5ng unfavourable condi5ons for an eclipse. It should be noted that under these circumstances the New Moon can either be above or below the line joining the Earth and the Sun due to the Moon’s 5lted orbit. The lower diagram shows condi5ons that are favourable for an eclipse (either Lunar or Solar) with the New/Full Moon coinciding with a 5me when the Moon is at one of the nodes. A Lunar eclipse occurs when the Moon enters the Earth’s shadow. A Solar eclipse occurs when the Moon’s shadow falls on the surface of the Earth. A total lunar eclipse occurs when the whole of the Moon passes through the umbra of the Earth’s shadow. Note that the cross-­‐sec5onal area of the umbra is larger than the Moon, so a total lunar eclipse is visible at all loca5ons on the night-­‐side of the Earth. A par5al eclipse occurs when only part of the Moon passes through the umbra. The fact that the shadow of the Earth cast on the Moon is curved provided Aristotle with evidence that the Earth is spherical. Even in totality the Moon does not disappear completely, but instead turns a red-­‐orange colour. Why is this ? A total solar eclipse occurs when the umbra of the Moon’s shadow reaches the surface of the Earth. The umbra has a maximum width of approximately 270 miles, so the path of totality forms a narrow band that runs across the Earth’s surface. During a total solar eclipse the very hot (T ~ 106 K) and diffuse outer layers of the Sun (the corona) become visible as the glare of the solar disc is blocked out by the Moon. An annular solar eclipse occurs when the umbra of the Moon’s shadow does not extend to the surface of the Earth. This arises because the Moon’s orbit around the Earth is not circular, and annular eclipses arise when the Moon is more distant from the Earth such that its apparent size on the sky is smaller. An observer located in the path of totality sees that the Moon does not quite cover the Sun’s disc at the moment of maximum eclipse, allowing a narrow annulus of the Sun to be visible. A par5al solar eclipse occurs for observers who are located in the penumbra of the Moon’s shadow. For these observers only a frac5on of the Sun is covered by the Moon. Measuring sizes and distances in astronomy using triangula5on Measuring the size of the Earth Around 200 B.C. the Greek astronomer Eratosthenes measured the size of the Earth. It was known that on the summer sols5ce the Sun was directly overhead in the town of Syene (near Aswan in Egypt), such that it shone directly into deep water wells. In Alexandria the Sun at local noon was measured to be 7o from the zenith. Eratosthenes realised that the curvature of the Earth was responsible for this observed difference in solar posi5on in the sky. Using the diagram on the right it is clear that the following rela5on holds: 7o/360o = (Distance between Alexandria and Syene)/(Earth’s circumference) Alexandria was known to be 5000 stades (1 stade is believed to be ~1/6 km) north of Syene, giving a value of 42,000 km for the Earth’s circumference. This is very close to the real value 40,000 km Measuring Earth, Moon, Sun distances Around 280 BC Aristarchus of Samos proposed a method for measuring the rela5ve distance between the Sun and Moon. At first or third quarter Moon, Aristarchus knew that the Sun, Moon and Earth form a right angle. He es5mated that the angle between Moon and Sun at first or third quarter is 87o, giving the remaining angle of the triangle at the Sun as 3o. Trigonometry tells us that for the right-­‐angle triangle shown in the diagram Sin(3o) = (Earth-­‐Moon distance)/(Earth-­‐Sun distance) giving an es5mated value of (Earth-­‐Moon distance)/(Earth-­‐Sun distance) ~ 1/20 The true value is ~ 1/380. Aristarchus was inaccurate in his measurement of the Moon-­‐Earth-­‐Sun angle, leading to this error. Aristarchus also devised a method of measuring the rela5ve sizes of the Earth and Moon during a total lunar eclipse. He combined this measurement with the fact that the Sun and Moon appear to be the same size during a total solar eclipse, and with his previous measurement of the ra5o of the Earth-­‐Moon and Earth-­‐Sun distances, to es5mate the rela5ve size of the Earth and Sun. Aristarchus goes down as the first person in wriNen history to propose a heliocentric model for the Solar System. See coursework 1 for further details. The upper diagram shows that a larger object further away from an observer has the same angular size (or subtends the same angle) as a smaller object close by. This is approximately the situa5on with the Sun and the Moon as viewed from the Earth during a total solar eclipse. The boNom image shows that we can use the small angle formula to relate the angular size α of a distant object to its distance d and its diameter D through the expression α = d/D. Measuring stellar distances using trigonometric parallax Trigonometric parallax is a method used by astronomers for measuring distances to stars. Consider the diagram below that shows observa5ons of a nearby star taken 6 months apart when the Earth is on opposite sides of its orbit. This is the largest baseline available to astronomers when measuring astronomical distances using parallax. The projected posi5on of the nearby star measured against the background of much more distant stars shiYs when the two observa5ons are compared. This is shown in the lower diagram where we we see that the apparent posi5on of the yellow star has shiYed. This shiY of a nearby object against a background of more distant objects when viewed from different posi5ons is called parallax. The upper diagram shows that the angular shiY in the posi5on of the nearby star when viewed 6 months apart equals 2p, where p is called the parallax angle. The mean radius of the Earth’s orbit is 1 Astronomical Unit (AU) which equals 1.5 x 1011 m. Measuring the distance to the star, d, in AU and the parallax angle in radians gives the rela5on Tan(p) = 1/d. The angle p is always very small, allowing us to write p = 1/d. We now introduce a unit of distance called the parsec, defined such that a star whose parallax angle p is equal to 1 arcsecond has a distance equal to 1 parsec. Expressed in radians, 1 arcsecond = 1/206264.8 making 1 parsec equal to 206264.8 AU or 3.26 light years. If we express the parallax angle, p, in units of arcseconds and the distance, d, in parsecs we have the simple rela5on p=1/d. A star with a parallax angle of 1 arcsecond (1”) has a distance equal to one parsec for a basesline of 1 AU. The first successful measurement of stellar parallax was conducted by Friedrich Bessel in 1839 for the star 61 Cygni, which has a parallax angle p=0.28”. The Sun’s closest stellar neighbour Proxima Centauri has a parallax angle p=0.78”.