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The size of the solar system • The size of Earth, Moon and the Sun • Distances to SS objects • Kepler’s laws and elliptical orbits The Solar System • Remarkably, with a few careful observations it is possible to measure the scale of the solar system Size and shape of Earth • The Earth has been known to be spherical since the time of the early Greeks. Some of the evidence in favour of this was: Size and shape of Earth • The Earth has been known to be spherical since the time of the early Greeks. Some of the evidence in favour of this was: 1. at sea, land at sea level disappears before hills; hulls of ships at sea vanish before their masts Size and shape of Earth • The Earth has been known to be spherical since the time of the early Greeks. Some of the evidence in favour of this was: 1. at sea, land at sea level disappears before hills; hulls of ships at sea vanish before their masts 2. the altitude of stars in the sky depends on how far north or south the observer is Size and shape of Earth • The Earth has been known to be spherical since the time of the early Greeks. Some of the evidence in favour of this was: 1. at sea, land at sea level disappears before hills; hulls of ships at sea vanish before their masts 2. the altitude of stars in the sky depends on how far north or south the observer is 3. in lunar eclipses (Earth passing between Sun and Moon) the shadow is always circular Size and Shape of Earth • Eratosthenes used the assumption of a spherical Earth and his observation of the difference of altitude of the Sun at Syene (directly overhead on a known date) and at Alexandria, 5000 stadia farther north. • At Alexandria the Sun was 7.2° north of overhead • Using basic geometry he related the distance on the spherical Earth (5000 stadia) to the angular distance around the circumference (7.2°) and found a radius of ~39000 stadia. • We don’t know exactly what this unit is in our terms, but best estimates suggest 1 stadium = 157 m; with this conversion Eratosthenes’ method gives a radius for Earth of ~6250km. This is very close to the modern value of 6378km. The Moon: Eclipses Eclipses occur when the Moon comes between the Earth and Sun. Provides clear evidence that Moon is closer than Sun Solar Eclipses • Eclipses are so spectacular because of the purely coincidental fact that the moon and Sun have similar angular sizes Lunar Eclipses Lunar eclipses occur when Earth blocks sunlight to the Moon Lunar eclipses always have rounded edge: further evidence that Earth is spherical. Distance to the Moon Lunar eclipses can be used to determine distance to the Moon • Angular diameter of the Sun is 0.53 degrees • Knowing Earth’s diameter (13,000 km) you can find the extent of Earth’s shadow: 1.4 million km. • From observing the radius of curvature of the shadow we see the angular size of Earth’s shadow at the distance of the Moon is about 1.5 degrees. • Can use geometry to show distance to Moon is about 350,0000 km Given the angular size of the moon (0.5 deg) and its distance of 350,000 km we can find its size. Distance to the Sun • Aristarchos observed the angle between the Moon and Sun at quarter phase; this told him the relative distances of Sun and Moon. He measured this angle to be 87 degrees. The modern value is 89.75 degrees. (Why is this measurement hard?) Sun is about 400 times farther away than Moon Since Sun and Moon have the same apparent diameter when viewed from Earth, the Sun must also be 400 times larger than the Moon Planetary motions • The planets move relative to the background stars. • Sometimes they show complex retrograde motions • Skygazer demonstration Epicycles • Epicycles were introduced to explain the non-uniform velocities of planets, in a geocentric, circular-orbit theory Retrograde motion • Retrograde motion is a natural outcome of the heliocentric model • Inner planets orbit more quickly than outer planets, and so “overtake” them Distances to Interior planets • Venus and Mercury follow the Sun around the ecliptic: means their orbits are smaller than Earth’s • At greatest elongation a line between the Sun and planet is perpendicular to a line between Earth and planet. rPlanet Sun sin rEarth Sun • E.g. for Venus, =46 degrees, so the distance from Venus to the Sun is 0.72 times the EarthSun distance Distances to exterior planets • Exterior planets can be found anywhere in the zodiacal belt • The true orbital period of the planet (sidereal period) tells how long it takes the planet to return to point P. • Observe the angles PES(initially) and PES (one superior planet period later). • The angle ESE’ is known from the Earth’s orbital period vs. the planets. And the triangles can be solved. Break Tycho Brahe • Brahe (1546-1601) believed in a geocentric Universe: the Sun and moon go around the Earth (but the other planets go around the Sun) • However, he also believed that this theory could be tested by making sufficiently accurate observations At time this was a revolutionary approach: different from the idea that phenomena could be understood through philosophical discourse alone Arguably the first application of the scientific method Tycho Brahe’s observations • Made very accurate, naked eye observations of planetary motion Used devices for measuring angles and positions To measure time, he used the planetary motions themselves. Clocks were rare and the pendulum clock had not been invented sextant wall quadrant Kepler’s Laws Johannes Kepler derived the following 3 empirical laws, based on Tycho Brahe’s careful observations of planetary positions (astrometry). 1. A planet orbits the Sun in an ellipse, with the Sun at one focus (supporting the Copernican heliocentric model and disproving Brahe’s hypothesis) 2. A line connecting a planet to the Sun sweeps out equal areas in equal time intervals 3. PP2 2 =aa33, where P is the period and a is the average distance from the Sun. What is an ellipse? Definition: An ellipse is a closed curve defined by the locus of all points such that the sum of the distances from the two foci is a constant: r r 2a Ellipticity: Relates the semi-major (a) and semi-minor (b) axes: a 2 a 2e 2 b 2 b 1 e2 a Equation of an ellipse: r 2 r 2 sin 2 2ae r cos 2 Substituting r r 2a and rearranging we get: a 1 e2 r 1 e cos Ellipses Calculate the aphelion and perihelion distances for Halley’s comet, which has a semi-major axis of 17.9 AU and an eccentricity of 0.967. Kepler’s Second Law 2. A line connecting a planet to the Sun sweeps out equal areas in equal time intervals This is just a consequence of angular momentum conservation. L r p mrv zˆ Angular momentum conservation Since L is constant, La L p (aphelion=perihilion) mra va mrp v p va rp 1 e v p ra 1 e Angular momentum conservation How much faster does Earth move at perihelion compared with aphelion? The eccentricity is e=0.0167 vp va 1 e 1 e 1.0167 0.9833 1.034 i.e. 3.4% faster Orbital angular momentum We know the angular momentum is constant; but what is its value? Lrp rv zˆ Since L is constant, we can take A and t at any time, or over any time interval. dA L 2m dt L 2m 2m Aellipse P a 2 1 e 2 P Kepler’s Third Law The general form of Kepler’s third law can be derived from Newton’s laws. 4 2 a 3 P G ( M m) 2 Circular Velocity • A body in circular motion will have a constant velocity determined by the force it must “balance” to stay in orbit. • By equating the circular acceleration and the acceleration of a mass due to gravity: vcirc GM r • where M is the mass of the central body and r is the separation between the orbiting body and the central mass. Escape velocity • Escape velocity is the velocity a mass must have to escape the gravitational pull of the mass to which it is “attracted”. • We define a mass as being able to escape if it can move to an infinite distance just when its velocity reaches zero. At this point its net energy is zero and so we have: GMm 1 2 mvesc r 2 vesc 2GM r Escape velocity What is the escape velocity at a) the surface of the Earth? b) the surface of the asteroid Ceres? Orbital Energy GMm m 2 GMm E v 2a 2 r • In the solar system we observe bodies of all orbital types: planets etc. = elliptical, some nearly circular; comets = elliptical, parabolic, hyperbolic; some like comets or miscellaneous debris have low energy orbits and we see them plunging into the Sun or other bodies orbit type v Etot e circular v=vcirc E<0 e=0 elliptical vcirc<v<vesc E<0 0<e<1 parabolic v=vesc E= 0 e=1 hyperbolic v>vesc E>0 e>1 Vis-Viva Equation • Since we know the relation between orbital energy, distance, and velocity we can find a general formula which relates them all – the Vis Viva equation 1 1 v (r ) 2GM r 2a 2 • It is derived by integrating the equation for total energy as a function of distance and incorporating the assumption that total orbital energy is constant no matter what the distance from the centre of mass. • This powerful equation does not depend on orbital eccentricity. • For instance, if we observe a new object in the SS and know its current velocity and distance, we can determine its orbital semimajor axis and thus have some idea where it came from. Vis-viva equation A meteor is observed to be traveling at a velocity of 42 km/s as it hits the Earth’s atmosphere. Where did it come from? Next Lecture Physical processes in the SS Tidal forces Resonances Solar Wind