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Last Week. Objectives of Telescope design. - Light Grasp - resolving power - Angular magnification. 2. Refracting and reflecting telescopes. 3. Aberrations and remedies 4. Effects of the atmosphere - adaptive optics
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Last Week. • Objectives of Telescope design. • - Light Grasp • - resolving power • - Angular magnification. • 2. Refracting and reflecting telescopes. • 3. Aberrations and remedies • 4. Effects of the atmosphere • - adaptive optics • 5. Telescopes for other parts of the spectrum. • - radiotelescopes and arrays of • radiotelescopes. R = m = 1.22/b where m is the angular separation of the two objects and b is the telescope aperture. Rayleigh’s criterion
Topics related to Gravity 1. Kepler’s Laws 2. Artificial Satellites 2. Escape Velocity 3. Tides 4. Synchronous orbits 5. Planetary ring systems 6.The Roche Limit 7.Masses of stars in a binary system
Starting Point – Kepler’s Laws. • Tycho Brahe (1546 – 1601) From 1576-1597 made detailed measurements of the positions of the planets with an accuracy of 1 arcminute. He did not believe that Earth orbited the Sun because he could not see “local” stars moving relative to the distant background of stars. Johannes Kepler (1571-1630) He used Brahe’s measurements to create a model of planetary motion. He broke with tradition-it had been believed that heavenly bodies move in circles. Kepler explained planetary motions in terms of the planets moving in ellipses.
The Ellipse y • Defn - Locus of all points the sum of whose distances from two fixed points (foci) is a constant. (0,+b) (-f,0) (+f,0) (-a,0) (+a,0) •Position of foci (a + f) + (a-f) = (f2 + b2)1/2 + (f2 + b2)1/2 i.e. f = (a2 - b2)1/2 x (0,-b) • Eccentricity e = f = (a2 - b2)1/2 = [1 - b2 / a2]1/2 [ Definition] a a • When e = 0, a = b and we have a circle i.e. x2 + y2 = r2
Kepler’s Laws (1618) 1.The planets orbit the Sun in ellipses with the Sun at one focus. 2.The line joining the Sun and a planet sweeps out equal areas in equal times. 3.The square of the period of a planet is proportional to the cube of the semi-major axis of the ellipse. P2 a3 Note:- Elliptical orbits were an essential innovation but for simple calculations one can assume that the orbits are circles. In general it is a good approximation. Convenient Measure of distance -Astronomical Unit(1 au) = Average Earth-Sun distance = 1.496 x 1013 cm = 1.496 x 1011 m
Measures of Distance – So Far 1. Convenient Measure of distance -Astronomical Unit(1 au) = Average Earth-Sun distance = 1.496 x 1013 cm = 1.496 x 1011 m = 1.496 x 108 km. 2. LIGHT YEAR (ly) is the distance travelled by light in vacuum in 1 year. 1 ly = 9.461 x 1012 km Note :- 1ly = 6.324 x 104 au
Kepler’s Third Law Plot of a3 versus P2 for the planets in the Solar system - Here a is in AU and P is in Earth Years. a3 Clearly P2 a3 P2 All three of Kepler’s Laws are rigorously obeyed wherever two objects move under their mutual gravitational attraction.
Planetary Orbits Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Semi-major axis (106 km) 57.9 108.2 149.6 227.9 778.4 1424 2871 4499 5906 Sidereal Period 0.241 0.615 1.0 1.88 11.9 29.5 84.0 165 248 Orbital Eccent. 0.21 0.01 0.02 0.09 0.05 0.06 0.05 0.01 0.25 Direction of revn All in same direction Angle to plane of ecliptic(degs.) 7.0 3.4 0 1.8 1.3 2.5 0.8 1.8 17.1 Angle of Plane to spin axis(degs) 0.1 178 23.5 25.2 3.1 26.7 97.9 29.6 122 Rotation Period (Days) 58.7 243 1.0 1.03 0.41 0.43 0.72 0.67 6.4 Surface Temp. 1-700 730 300 220 130 97 58 58 50 Mass(1024 kgm) 0.33 4.87 5.98 0.64 1900 569 86.8 102 0.013
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto All the same Direction of revolution Direction of rotation on axis Same Opp. Same Same Same Same Sideways Same Sideways All these characteristics have to be explained by any model of the formation of the solar system.
Reminder of Newton’s Laws 1.A body remains at rest, or moves in a straight line at constant speed, unless acted on by an external force. 2.The acceleration of an object is proportional to the force acting on the object. F = m = d(mv) dt 3.Whenever a body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.
Directon of force to keep Earth in circular orbit Sun •At any instant, direction of motion is tangential to the orbit. •Centripetal force(FCP) acts towards the Sun to keep it in circular orbit. FCP = mv2 Direction of motion r Newton reasoned that if Sun attracts Earth then Earth attracts Sun F =G.m1.m2 [Universal Law of Gravitation] r2
The Effect of Centre-of-Mass Centre-of-mass dm dM r The centre-of-mass is given by dM = m . r ( m + M ) m M dm = M . r ( m + M ) If m/M 0.01 we cannot assume that we have a fixed, central body. If so we can no longer simplify our equations and we have to take account of the fact that the motion is about the centre-of-mass.
The Effect of Centre-of-Mass P2 = 42.r3 = 42.(M + m)2 .dM3 Mv2 = G.M.m 1. dM r2 Now the period P for an orbit is 2.radius/velocity i.e. v = 2. dM P 2. Mv2 = M. (2. dM/ P)2 = G.M.m r2 Since dM = m .r (M + m) dM dM = 42.(M + m)2.dm3 G.m3 G.(M + m) G.M3 3. In the case of the Earth and Sun the imbalance in mass is such that the approximation of a static Sun is a good one. However Kepler’s Laws apply to all systems moving under gravity and often the masses are closer together so we cannot assume that one of them is fixed. [ Mass of Sun = 1.99 x 1030 kg Mass of Earth = 5.97 x 1024kg ]
Artificial Satellites 1.There are many possible orbits for our satellites depending on their function.One useful orbit is the GEOSTATIONARY orbit. i.e.orbit keeps the satellite in the same place relative to the Earth. 2.Clear condition is that satellite orbits with same period and direction as the Earth. Hence P2 = 42.r3 r3 = G.M.P2 G.M 42 = 6.67.10-11.6.10+24(24.60.60)2 42 r = 42,300 km. So our satellite orbits at about 5.6 Earth Radii above the surface. Note that this is independent of the mass of the satellite. [Earth’s radius is 6370 km.]
Escape Velocity 1.There are many reasons why this is important. It is the velocity required to escape from the gravitational field of another body. This is the velocity needed to take it to infinity. 2.To take the object to infinity it is necessary to do work against gravity. Work done = F.dr = G.M.m.dr with the integral from r to r2 = G.M.m where rp is the radius of the planet or other large body. rp 3.This is the kinetic energy the projectile must have initially to escape to infinity. 1.mv2escape =GMm/rp For the Earth vescape = [2x6.67x10-11x 5.98x1024/6378x103]1/2 11.2 kms-1 2
Ocean Tides • Tidal forces are due to differential forces across a body.They act on oceans and on the Earth itself. • Tides are closely connected with the position of the Moon. AccelN due to Moon is greater at A than B,which is greater than at C. Water is fluid & it drops towards Moon most quickly at A and at C it is left behind.At A and C we have HIGH TIDE and at B LOW TIDE. • As the Earth spins on its axis( Moon’s period is 27 times longer ) every longitude on Earth will experience two high tides and two low tides per day. The bulge(tide) is always aimed slightly ahead of the Moon.
Ocean Tides As the Earth spins on its axis( Moon’s period is 27 times longer ) every longitude on Earth will experience two high tides and two low tides per day. They occur every 25 hours. The bulge (tide) is always aimed slightly ahead of the line joining the Earth and Moon. There are many local effects and in some places there is only one high tide a day.
The Moon versus the Sun • FS MS.rM2 2 x 1030 ( 3.84 x 108 )2 170 = = FM MM.rS2 7.4 x 1022( 150 x 109 )2 HOWEVER • F = -2GMm so the comparative effect of the Sun and Moon r3 r is given by FS MS . rM3 0.465 = FM rS3 . MM • So tide-raising ability of Moon is 2.15 times effect of Sun but effect of Sun is not negligible. • Effects of Sun and Moon combine linearly and vectorially. When Sun and Moon are aligned (Full or New Moon) we get a maximum tide 1.465 times normal.These are Spring tides(every 2 weeks).At Half Moon(at right angles) tides are smaller-Neap Tides.
Synchronous Orbits a)The gravitational forces on Earth due to the Moon. b)The differential forces at the same points. 1.Normally we think of tides in terms of the Oceans but there are effects on solid objects such as the Earth or Moon.Thus the Earth feels different forces on its two sides.The differential forces elongate the shape of the Earth,which behaves like a fluid.The result is bulges of approx 10 cm.along the line joining the centres of mass. 2.Again Earth’s tidal bulges are not aligned with the Moon. Frictional forces on the surface drag the bulge axis ahead of the line joining the Earth-Moon. Friction is a dissipative force so rotational K.E. is constantly being lost and Earth spins more slowly[1.6 cms per century].The Moon is also drifting away at 3-4 cm per year.
Synchronous Orbits 1.How are these things related? 2.Consider torques on the Earth due to the bulges. Bulge A leads the Moon and is closer to it.Force on A is greater than on B.The result is a net torque slowing down the Earth.It reduces angular momentum.However A.M. is conserved in this isolated system. 3.At the same time Bulge A is pulling the Moon forward,speeding the satellite up and causing it to move further away (mvr = const.). Note:-We are assuming the influence of planets and Sun are negligible 4.Because of these tidal effects the Earth will eventually slow down so that the same side of the Earth will always face the Sun in the same way that the Moon faces the Earth. In the past the Moon would have been much closer to the Earth with a period as short as a week. This is a more general phenomenon.
The Roche Limit 1.Edouard Roche(1820-1883).-When a comet or Moon or other large object is spiralling in on a planet (mass M) it is unlikely to collide with it. Instead it breaks up at the ROCHE LIMIT. 2.As the object approaches it has a differential force across it dF = -2G.M - the differential force on unit mass. dr r3 3.Tidal forces increase rapidly as distance decreases.the result is that the shape of the object[Moon,comet,asteroid etc]becomes elongated. 4.If we assume that the object is a fluid, then when r is small it becomes impossible to define a shape where the force of gravity is always at right angles to the surface.Then the material will always flow in the direction of the net gravitational force. 5.If oscillations start then the object will break up under tidal disruption.
The Roche Limit 1.Roche Limit = Max. orbital radius for which tidal disruption occurs. Assume disruption occurs when diff. grav. force exceeds “self” grav. force holding it together and that “moon” and planet are spherical. We neglect centrifugal effects. For disruption, GM.m < 2GMR , where R is the radius of the “moon”, r is the distance between the “moon” and the planet and M is the mass of the planet. r3 R2 2.Now M = 4/3..Rp3p and m = 4/3..Rm3.m 3.Solving for r we find r < 21/3(p / m)1/3Rp Roche found 2.456 4.Examples:-Comet Shoemaker-Levy Rings of Saturn
Shoemaker-Levy Comet captured into an orbit round Jupiter.Its orbit took it 1/3AU from Jupiter and Sun modified orbit so that we knew it would collide with the planet in July 1994. The comet broke up because it exceeded the Roche limit and the fragments crashed on the planet between July 16-24 Here we see the effect of the impacts as observed with the HST in the UV part of the spectrum.
Saturn’s Rings seen by Hubble Space Telescope at different wavelengths.
Physics of Atmospheres 1.The composition of the atmosphere of a planet is related to the simple idea of escape velocity.[vESCAPE = [2G.M/rP]1/2 =11.2 kms-1 for Earth] 2.The Temperature is a key parameter in the formation and evolution of a planet. Inter alia it affects the nature of the atmosphere. 3.For a star L = 4.R2..T4 where we have assumed a spherical star of radius R 4.Under equilm conditions the total energy content of a planet is constant. Assume planet is a black-body of radius RP and temp.TP in a circular orbit at distance D from the star.Assume the planet reflects a fraction of incoming light[the ALBEDO] 5.It can be shown that TP = TS(1-)1/4[RS/2D]1/2, i.e. TP is proportional to the surface temp. of the star and does not depend on the size of the planet. 6.Using = 0.3 the temp. of the Earth is 255K.This is substantially below the value of 300K.This is because it ignores the Greenhouse effect.This is a significant warming due largely to water vapour.
Physics of Atmospheres 7.From Wien’s Law MAX.T = 2.9 x 10-3 m.K For Earth MAX = 1000 nm. Hence it emits in the infrared. This infrared radiation is absorbed and re-emitted by the atmospheric greenhouse gases( CO2,CH4,and chlorofluorocarbons and H2O vapour) This acts as a thermal blanket and raises T by about 34 degrees. [This is the reason for the concern about increases in CO2 emissions] 8.Evolution of the atmosphere depends on T of solar nebula during formation together with planet’s temp.,gravity,composition,&chemistry following formation. Later outgassing from rocks and volcanoes plays a part. 9. Figure shows the Maxwell- Boltzmann distribution for the velocities of molecules or atoms in a gas. Some will exceed the escape velocity.
Law of Gravity as a function of Distance/Time • Cavendish showed in the lab. that F = G.M1.M2 r2 Where G = 6.672.10-11 Nm2kg-2 • Earth ,Sun etc are spherical-What we expect for aggregation of material.[Note:-it is oblate spheroid.] • Moons of Jupiter-originally they appeared to vary in orbital period. They were ahead(behind) time when close(far) from Earth.this is effect of finite velocity of light. • Analysis of motions of Jupiter ,Saturn and Uranus in terms of grav. Force.First two worked but not third.Adams and Leverrier independently suggested another planet- Pluto It works on scale of Solar System
Law of Gravity with Distance/Time • Binary stars(majority)-example SIRIUS-8.6ly away. • Picture shows measurements of relative positions of Sirius A&B -Perfect Ellipse but Sirius A is not at focus. This is because we see the elliptical orbit at an angle. • Globular Clusters are spherical
Coma Cluster - 20Mly across containing thousands of galaxies(300 seen here). It is about 270 Mly away.Two supergiant galaxies seen in centre.